Synchronous belt and pulley drive

ABSTRACT

A synchronous belt and pulley drive in which the drive between spaced pulleys is primarily by frictional contact of a belt on the pulley peripheries; synchronization is insured by providing spaced teeth on the belt which teeth are accommodated by tooth gaps in the periphery of the pulleys. The drive is further characterized by matching the pitch of the driveR pulley with the belt pitch under a first tension and matching the pitch of the driveN pulley with the belt pitch under a second tension, wherein the first tension is different and usually greater than the second tension.

BACKGROUND OF THE INVENTION

Current design practice for designing synchronous belt drives followsthe same general principles used for designing inverted tooth chaindrives wherein the chain (or belt, as the case may be) teeth carry theload imposed on the drive. However, a belt differs from a chain in atleast two important respects, i.e., belts, because of their constructionof elastomeric material, usually with a reinforcing cord and/or clothcovering, elongate much more than chains under load; and the resilientbelt teeth deflect much more than the relatively rigid teeth of a chain.In a belt drive, friction between the belt and the pulley peripheriescan be utilized to carry a major portion of the load.

The Invention

This invention relates to synchronous belt drives as, for example, thosein automotive timing arrangements. Other uses will be apparent to thoseskilled in the art.

One of the primary purposes of this invention is, in a drive using atoothed belt and toothed pulleys, to take advantage of belt-pulleyfriction as the primary load-carrying means. The principal function ofthe teeth is to eliminate excessive slip and insure and maintainsynchronization between the pulleys. By doing so, belt tooth deflectionand wear are minimized. This, in effect, transforms the toothed pulleysinto pulleys having a "variable" pitch. The teeth on the "tight" strandof belt (from the driveN to the driveR pulley) cannot interfere with thepulley teeth because the pulley tooth gaps are larger in depth andlength than the teeth of the belt, thus minimizing belt toothdeformation and wear. Therefore, because of the relatively small loadson the belt teeth, they can be spaced further apart, (the belt pitchextended) than in the normal, prior art toothed belt-sprocket drive. Forexample, a belt according to this invention may have approximatelyone-third (or less) the number of teeth than a conventional toothedbelt, both of which are usable for the same purpose. Because of thefewer teeth on the belts, fewer teeth or tooth gaps are necessary on thepulleys. The fewer gaps generally mean lower manufacturing costs. Byreducing the number of tooth gaps in the pulleys without increasing thesize of the gaps, the area of contact between the outer periphery of thepulleys and the belt is increased, as compared to prior art drives.Thus, the friction effect between the belt and the pulleys is enhanced.

One of the important aspects of the invention is that the pitch of thedriveR pulley substantially matches the belt pitch under a first tensionand the pitch of the driveN pulley substantially matches the belt pitchunder a second tension. The second tension is less than the firsttension and equals the "slack" side tension of the belt while the firsttension equals the tight or "taut" side tension of the belt. Thisrelationship will be more fully discussed in the detailed description ofthe invention.

THE DRAWINGS

FIG. 1 of the drawings illustrates a side elevational view of a driveconstructed to this invention;

FIG. 2 is a graph in which elastic elongation is plotted against beltload;

FIG. 3 illustrates schematically a drive according to this invention inwhich a belt tensioner is used; and

FIG. 4 is a schematic illustration of a typical drive according to thisinvention and marked for reference to the calculations in thespecification.

DETAILED DESCRIPTION

Looking at the drawings and especially FIG. 1, there is shown a beltdrive comprising a driveR pulley 10 and a driveN pulley 12, eachrotatable about its center and in the direction indicated by the arrows.The pulleys 10 and 12 are connected by a flexible belt 14 having aplurality of spaced teeth 16. The pulleys 10 and 12 are each constructedwith tooth gaps 18 and 20, respectively, each of which will accommodatea belt tooth at the proper time. Each tooth gap 18 and 20 has a depthand length greater than the depth and length of a belt tooth 16.

When the pulleys are rotating, the belt is subjected to a first tention(T₁) at the tight or taut side, i.e., from the driveN pulley 12 to thedriveR pulley 10, and to a tension (T₂) at the slack side, i.e., fromthe driveR pulley 10 to the driveN pulley 12. The pitch of the driveRpulley 10 approximately matches the pitch of the belt under tension T₁,while the pitch of the driveN pulley 12 approximately matches the pitchof the belt under tension T₂, which means that the pitch of the driveRpulley 10 differs from the pitch of the driveN pulley 12.

To design a belt drive based on the principles of this invention, thepitches of the pulleys are determined by the pitch of the belt strandentering the pulley. It is therefore necessary to know the elasticelongation--load characteristics of the belt. A reasonable size for thepitch diameter of the smallest pulley is selected, and the number oftooth gaps is chosen such that only two or three are provided toaccommodate the belt teeth; the lengths of the tooth gaps being one halfto one quarter of the spacing therebetween.

The total number of equally spaced tooth gaps (pitches) around theperiphery of the smallest pulley is (illustrated for example, in FIG. 1as the driveR pulley) such that only two or three pulley tooth gapsaccommodate the teeth in the portion of the belt wrapping the pulley.Pulleys having five to eight tooth gaps are feasible, depending on thewrap. The number of pitches (tooth gaps) in the other pulleys isgoverned by the speed ratios required.

Having selected a reasonable pitch diameter (d) for the smallest pulleyand the number of tooth gaps thereon (n), an approximate pitch (p) forthe drive (pulleys and belt) can now be determined by dividing thechosen number of tooth gaps into the selected pulley and pitchcircumference, i.e.

    p=πd/n.

When only two pulleys (besides an idler if necessary or desirable) areinvolved, the number of pitches in the belt can be determined byreference to Center Distance Factor Tables, e.g. Catalog 189 (1969)published by Uniroyal, Inc. In as much as such tables list centerdistances in terms of pitches, it is necessary to convert the givencenter distance (in inches) into pitches by dividing it by theapproximate pitch (p). The number of pitches in the belt is chosen tomake the required center distance (in pitches, determined above) matchas closely as possible a listed center distance value in the tables.Dividing the latter value (in pitches) into the given center distance(in inches) obtains a more exact value for belt pitch, hereinafterreferred to as "assumed" belt pitch.

Since the pitch of the pulleys should match the pitch of the beltstrands entering them, the pulley pitches are equal to the belt pitchplus the elastic pitch elongation of the entering belt strand due to theload on it. The pulley pitches will therefore differ from each other andfrom the "assumed" belt pitch. The linear length of belt required can befigured and compared with the actual belt length based on the number ofpitches and the assumed belt pitch. If the actual belt length exceedsthe calculated wrapped belt length, the calculations is repeated using asmaller assumed belt pitch (and vice-versa). Usually, not more thanthree trial calculations are necessary to obtain a correct solution.

FIG. 1 illustrates the relationship of the belt teeth and pulley teeth.For example, all the tooth gaps 18 or 20 have the same length; thelength of the belt teeth 16 are approximately three-fourths of thelength of the tooth gaps. The designations A to H represent the entrywall of the tooth gaps 20, from belt entry to belt exit, on the driveNpulley 12, and A' to H' represent the exit wall of the tooth gaps 20from the belt entry to belt exit also on the driveN pulley 12.

There is no clearance at A and H'; if the belt tooth length isapproximately three-fourths of the tooth gaps length, then A'=H=approx.1/4 tooth gaps length; and A<B<C<D<E<F<G<H=1/2 tooth gap length, and

    A'>B'>C'>D'>E'>F'>G'>H'=0

It is apparent that a similar relationship exists with respect to thetooth gaps 18 of the driveR pulley 10 and the length of belt teeth 16.

FIG. 1 of the drawing illustrates a belt-pulley drive constructedaccording to this invention wherein the pulley pitches correspond withthe pitch of the belt strand entering it. The belt is wrapped around thepulleys; the tooth gaps in the pulleys are larger than the belt teeth,so that the belt can be mounted on the pulleys without interference. Ifan idler is used to apply an installed tension (as in FIG. 3), as isgenerally the case when the drive is used for automotive applications,the installed tension will move the belt slightly in a clockwisedirection (as viewed in the drawing) on the driveR pulley, andcontra-clockwise (as .[.viewd.]. .Iadd.viewed .Iaddend.in the drawing)on the driveN pulley, but not sufficient to make the belt pitch matchthat of the pulleys. When the drive is running under full load, theslack strand (from driveR pulley to driveN pulley) relaxes as the tautstrand (from driveN pulley to driveR pulley) stretches, so that theaverage pitch of the belt (i.e., engaging the pulleys) remainsapproximately unchanged. Since the belt tooth which is in workingcontact with a pulley tooth-gap wall is farthest away from theengagement point of the belt with the pulleys, the tooth load thereon isminimal, and, if necessary, deflection of this tooth will transfer loadto the next adjacent belt tooth. This occurs only under abnormal loadconditions, if at all.

The drive of this invention is not limited to any particular tooth form,nor is the invention limited to "extended" pitch belts (widely spacedteeth); it can be used with standard belts. However, it will be obviousthat extended pitch belts and their corresponding pulleys will result ina drive which is less costly to manufacture when compared to belts andpulleys having closely adjacent teeth and tooth gaps. Also, thefrictional drive characteristics are enhanced with the extended pitch.

The application of the design principles of the disclosure to anautomotive timing drive are illustrated by the following calculationsfor these given conditions:

1. Center distance between pulleys 13.159 inches.

2. Installed belt tension, 45 lbs./strand. [This is also the tension atwhich the belt length is to be measured; heating the engine to 180° F.increases the crank shaft-cam shaft center distance enough to increasethe belt tension by 25 lbs. per strand].

3. Belt 0.375 inches pitch×0.600 inches wide.

4. Net working load on the belt, 16 lbs.

5. Conventional design resulted in the following specifications:

Belt: 0.3758 inches pitch×0.600 inches×99 pitches long.

Pulleys: 19 and 38 teeth, pitch 0.3758 inches.

[Note that the belt not only has a unique number of teeth but also aspecial pitch, and there is nothing "standard" about any of the drivecomponents. The life of such a belt may be marginal, as the tooth wearprobably will be excessive].

Employing the principles disclosed in this invention, the designprocedure is as follows:

[In addition to the center distance between pulleys, the installed belttension, the belt pitch and width given above, it is necessary to knowthe elastic elongation--load curve, shown in FIG. 2.]

a. The belt width is accepted as 0.600 inches and for convenience, allloads are reduced to pounds per inch of width per strand:

Installed tension and measuring load=45/0.600=75 lbs. per inch perstrand

Tension when engine is heated to 180° F.=(45+25) 0.600=116.5 lbs. perinch per strand.

Working load=16/0.600=27 lbs. per inch per strand.

Taut strand tension=116.5+27/2=130 lbs./(inch)(strand).

Slack strand tension=116.5-27/2=103 lbs./(inch)(strand).

b. A reasonable size for the pitch diameter of the small pulley isselected, e.g. 2.25 inch=d.

c. An arbitrary number of teeth, is chosen e.g. 7 or 8=n.

d. Since the speed ratio selected is 2:1, the large pulley must have 14or 16 teeth=N.

e. The approximate pitch is then determined p=πd/n=2.25π/7=1.010 inches2.25π/8=0.884 inches

f. The center distance in pitches is calculated CD=13.16/1.010=13.03pitches 13.16/0.884=14.90 pitches

g. Refer to standard Center Distance (CD) Tables (Uniroyal et al) toobtain belt length, N_(b) in pitches,

1.For N-n=7, N_(b) -N=23, then CD=13.203 pitches.

2. For N-n=8, N_(b) -N=26, then CD=14,946 pitches.

[Both of the above are very close to the required center distances ascalculated in (f). Either might be selected, but it is sufficient forillustrative purposes to use only one, for example, CD=13.203 pitches].

h. A more exact pitch is calculated. [This may not be the finallyselected pitch because it assumes the belt and pulley pitches to bealike, hence further adjustment will be required in order to match thepulley pitches with that of the entering belt. N=14, n=7, N_(b) =37,p=13.159/13.203=0.9967 inches].

i. The corresponding pulley pitch diameter is d=(7) (0.9967)/π=2.221inches.

j. Inasmuch as the belt is to be mounted over the pulleys with the idlerretracted and without exerting any appreciable force on the belt, thebelt load will be zero. Note that the use of "installed" has beenavoided here, as the installed tension signifies the belt load to whichthe idler, if used, must be adjusted after wrapping the unloaded beltover the pulleys. Since the driveR pulley pitch should match that of thetaut strand of the belt (130 pounds per inch per strand) reference tothe elastic elongation--load curve shows a length increase of 0.00093inches/inch from the no load condition making the pitch of the driveRequal to the belt pitch plus 0.00093 inches per inch. Similarly, theslack strand of the belt entering the driveN pulley under a tension of103 pounds per inch per strand elongates 0.00076 inch per inch; hence,the pitch of the driveN pulley should exceed that of the belt by 0.00076inches per inch. To determine the pitch and length of belt required tosatisfy the mismatched condition, it is necessary to use a trial anderror method, and higher precision is demanded than that afforded by theslide rule which was adequate up to this point.

k. Start with a value of p somewhat larger than that found in (h) above,say p_(b) =1.000 inches where p_(b) denotes the belt pitch. Then p_(r)=1.000+0.00093 =1.00093, where p_(r) is the pitch of the driving pulleyhaving radius r; and P_(R) =1,000+0.00076=1.00076 inches, P_(R) definingthe pitch of the driveN pulley of radius R.

Calculate the radius of each pulley:

r=(1.00093)(7)/2π=1.115121 inches

R=(1.00076)(14)/2π=(2.229863/1.114742)=R-r

Note: See FIG. 4 for pictorial representation of drive to define radii,distances and angles.

sin α=(R-r)/CD=1.114742/13.159=0.084713α=4.8595°

Half lengths, straight strands=13.159 cos α=13.1117 inches

On small pulley=85.1405(1.115121π)/180=1/6570 inches

Large pulley=94.8585(2.229863π)/180=3.6918 inches

total=18.4605 inches

Belt length required=36.9210 inches

P_(b) =36.9210/37=0.99786 inches

Try P_(b) =0.9970 inches

P_(r) =0.99783 inches

P_(R) =0.99776 inches

r=b 1.111778 inches

R=2.223178

α=4.8449°

Belt length required=36.8890 inches Actual belt length=(0.9970)(37pitches)=36.8890 inches

It must be remembered that the above belt pitch of 0.9970 inches is atno load. Applying the measuring load of 75 pounds/(inch)(strand) wouldincrease the pitch by 0.00055 inches making p=0.99755 inches and thelength (0.99755)(37 pitches)=36.909 inches.

1. Since the belt pitch under no load is shorter than pulley pitches by0.00093 inches and 0.00076 inches in order to avoid interference whenmounting the belts on the pulley it is necessary to provide pulley toothgap clearance.

For the small pulley, four teeth (pitches) may engage the belt.

Clearance=(0.00093)(4)=0.0037 inches per tooth.

For the large pulley, 8 of its 14 teeth will engage the belt.

Clearance=0.00076(8)=0.0061 inches per tooth.

m. Now the complete specification can be written.

Belt Same section and tooth size as inches pitch.

Pitch=0.99755 inches width=0.600 inches

Length=37 pitches=36.909 inches at measuring load of 45 pounds perstrand.

Installed tension=45 pounds per strand.

Pulleys: Small: 7 teeth, pitch=0.99793 inches pitch dia=2,2236 inchestooth gap clearance=0.004 inches minimum

Large: 14 teeth, pitch=0.99776 inches pitch dia=4.4463 inches tooth gapclearance=0.006 inches minimum

Consider 2 pulleys, PA and PB, as illustrated in FIG. 3, connected by aflat belt. Pulley PC in the system is simply an idler, the function ofwhich is to apply an initial or installed tension on the belt. When noturning moment is applied to the driveR pulley PA, the tensions in thetwo parts of the belt are alike (except possible for friction of thebearings) and is due to the installed tension applied by idler pulleyPC.

It is evident that this initial will cause the belt to exert pressure onthe faces of the pulleys, and this pressure will induce a frictionalresistance opposing relative sliding between the belt and pulleys. If aturning moment is applied to PA and a resisting moment to PB, thefrictional resistance will increase the tension in the upper strand anddecrease the tension in the lower strand. Designate these tensions by T₁and T₂ respectively. These correspond to the tensions T₁ and T₂ in thedescription of FIG. 1. It is evident that the tendency of the belt toslip around the pulleys owing to the difference in tension on the twoparts of the belt is resisted by the frictional resistance between thebelt and pulley faces. The difference in tensions tend to rotate pulleyPB and when the turning movement (T₁ -T₂) r₁ becomes equal to theresisting moment applied to PB, rotation will take place, r₁ being theradius of the driveN pulley.

If the difference between T₁ and T₂, which is necessary to overcome theresisting moment is small compared to the frictional resistance betweenthe belt and pulleys, no slipping of the belt on the pulleys will occur.

In addition to the slipping action described above, all belts aresubject to what is known as "creep". Consider a piece of belt of unitlength moving onto the pulley PA under tension T₁. As this piece of beltof unit length moves around the pulley, the tension to which it issubjected decreased from T₁ to T₂, and owing to its elasticity, thepiece shrinks in length accordingly. The pulley PA, therefore,continually receives a greater length of belt than it delivers, and thevelocity of the pulley surface is faster than that of the belt movingover it. Similarly, pulley PB receives a lesser length of belt than itdelivers, and its surface velocity is slower than that of the beltmoving over it. This creeping of the belt as it moves over the pulleysresults in some unavoidable loss of power. The total loss of speed dueto both slip and creep should not exceed 3 percent; that is, the surfacespeed of the driveR pulley should not exceed that of the driveN pulleyby more than 3 percent. When it approaches 20 percent, there is dangerof the belt sliding off the pulley entirely.

When slipping is impending, the equation relating belt tensions T₁ andT₂ to the coefficient of friction, μ, and the angle of belt wrap, θ, inradians, is

    T.sub.1 =T.sub.2 e.sup.μθ

where e is the base of natural logarithms, 2.718. This neglects theeffect of centrifugal action, which is really not significant at beltvelocities below 2,000 feet per minute.

If T₁ /T₂ is less than or equal to e.sup.μθ, the belt will not slip onthe pulleys; for ratios larger than this, slipping will occur. In allcases, however, the belt will creep on the pulleys. As the value of T₁approaches that of T₂, (T₁ /T₂ →1), the amount of creep will diminishbecause there is less change in the length of a unit piece of beltmoving over the pulley. When T₁ =T₂, we have the condition "asinstalled" and no power can be transmitted.

By designing the drive so that T₁ /T₂ =e.sup.μθ, it is possible to getequal surface velocities at both pulleys if the radius of the driveRpulley is increased and radius of the driveN pulley is decreased tocompensate for the change in belt length in its passage around thepulleys. Assuming the belt is to be perfectly elastic, the elongation εin inches per inch can be expressed as:

    ε=K(T-T.sub.2)

and since T=T₂ e.sup.μθ

    ε=KT.sub.2 (e.sup.μθ -1)

and the change in the belt length (l₁ -l₂) wrapping θ radians on apulley of radius r is l₁ -l₂ =KT₂ r (.sub.μ¹ e.sup.μθ -θ--.sub.μ¹)

All of of the above concerns flat belts running over pulleys having flatfaces which are obviously capable of accommodating the belt properlyregardless of the fact that pieces of belt having unit length vary withthe change in belt tension therearound.

Now consider a synchronous belt drive in which the belt is provided withteeth to engage tooth gaps in the pulleys. The "unit length" referred toflat belts now becomes "pitch", p, for a toothed belt. Since the belt iselastic, in order to determine pitch it is necessary to measure thelength of the belt subjected to a specified load called the "measuringload" and divide this by the number of teeth in the belt. It is apparentthat the pitch or spacing between teeth will be greater or smallerdepending on whether the belt tension is larger or smaller than themeasuring load. Usually the measuring load is less than the installedbelt tension.

If the pulleys are designed to match the pitch of the belt at themeasuring load, they cannot accommodate the belt without interferencebetween the belt teeth and pulley tooth gaps when the drive is runningsince the belt pitch changes continuously around the pulley.Furthermore, if the installed tension differs from the measuring load,there is a belt pulley pitch mismatch when the drive is at rest.

It is apparent that the above conventional design must subject the beltto abrupt changes of pitch whenever the belt engages or disengages thesprockets. This is accommodated only by the elastic deformation of therelatively soft teeth on the belt during the interference with therelatively rigid pulley teeth.

The invention herein involves at least two, and preferably three,changes from the above practice. The first is to make the pulley pitchesmatch the pitch of the belt engaging or entering the pulley. From themaximum torque to be transmitted, the required driving force, F can becalculated. When the belt transmits power, the tension is increased onthe tight side and decreased on the slack side until the difference intension, T₁ -T₂, is equal to the required driving force, F. This isaccomplished by what virtually amounts to shortening the belt on thetight side, a given amount by transferring this amount to the slackside. If the relation between elongation and tension is linear over therange T₂ to T₁, then the increase in tension on the tight side willequal the decrease in tension on the slack side; in which case, T₁=T_(i) +F/2, and T₂ =T_(i) -F/2, where T_(i) is the installed tension.When the relation is non-linear, it is only necessary to choose T₁ andT₂ from the graph, FIG. 2 such that T₁ -T₂ =F and the elongation betweenT₁ and T_(i) is equal to the elongation between T_(i) and T₂. The pitchof the belt engaging each pulley can now be calculated. This determinesthe pitch of the pulleys, permitting the belt teeth to engage the pulleytooth gaps without interference because the belt and pulley pitches arealike on engagement.

The second change provides clearance in the pulley tooth gaps whichaccomplishes two purposes: (1) it permits the belt to wrap the pulleywithout interference between the belt teeth and the pulley tooth gapswhen the belt is mounted on the pulleys and when the installed tensionis applied by adjustment of the idler; and, more importantly, (2) itgives the belt freedom to change length (pitch) as the belt tensionchanges in traversing around the pulley, thereby taking advantage offriction, discussed earlier, as the principal means of carrying the beltload and avoiding slip. This frictional assistance significantly reducesthe load imposed on the belt teeth, the primary function of whichbecomes one of maintaining synchronization during load pulsations, speedfluctuations, etc. This feature, in essence, makes the toothed pulleysinto essentially flat pulleys. The pitch of the driveR pulley is largerthan that of the driveN pulley; a requirement which is necessary tocompensate for creep and make the surface velocities of both pulleysalike.

Finally, because of the reduced belt tooth loads resulting from theabove two design improvements, it is apparent that fewer teeth canprovide the synchronization, resulting in what might be called .[.and.]..Iadd.an .Iaddend."extended pitch" belt, wherein the belt tooth sizewould be that of a standard smaller pitch belt, but the spacing betweenthe teeth extended to something considerably larger, and not an integralnumber of the pitch defining the belt tooth size as is illustrated inFIG. 1. The procedure to follow is (a) assign an approximate diameterfor the smallest pulley and arbitrarily fix the number of teeth in eachpulley to meet the specified speed ratio; (b) calculate the approximatebelt pitch required to meet the center distance specification; (c)reduce these approximations to precise dimensions by trial and error.The elastic elongation- load characteristics of the belt must be known,and the minimum installed tension selected such that the ratio of belttensions, T₁ /T₂, at maximum drive load equals or exceeds e.sup.μθ wheree is the base for natural logarithms, 2.718, μ is the coefficient offriction between the belt and pulley face, and θ is the angle of wrap onthe pulley. The choice of this initial tension will not permit the beltto slip, so that friction alone should be able to carry the load.

I claim: .[.1. A synchronous belt and pulley drive comprising: and sidesof the tooth gaps. .Iaddend. .Iadd.
 13. A synchronous belt and pulleydrive comprising:a driver pulley and a driven pulley; said pulleys beingspaced from one another and each having bearing surfaces separatedcircumferentially by a plurality of uniformly spaced recesses foraccommodating belt teeth, an endless belt engaging the pulleys andhaving on its inner surface a plurality of spaced teeth, the spacingbetween adjacent teeth being uniform in the circumferential directionwhen the belt is free of tension, the drive between said respectivepulleys being primarily by friction between the belt and said bearingsurfaces, the relationship of belt teeth and pulley recesses insuringsynchronization of the drive, said pulley recesses beingcircumferentially wider and radially deeper than said belt teeth, thecircumferential extent of the bearing surfaces being substantiallygreater than the circumferential width of the recesses, thecircumferential extent of the bearing surfaces on the driver pulleybeing matched to the spacing between adjacent belt teeth when the beltis stretched under a first tension, and the circumferential extent ofthe bearing surfaces on the driven pulley being matched to the spacingbetween adjacent belt teeth when the belt is under a second anddifferent tension. .Iaddend.